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Calculate the gradient Vf of the following functions, f
Chapter 4, Problem 4.12(choose chapter or problem)
Calculate the gradient \(\nabla f\) of the following functions, \(f(x, y, z)\):
(a) \(f=x^{2}+z^{3}\).
(b) \(f=k y\), where k is a constant.
(c) \(f=r \equiv \sqrt{x^{2}+y^{2}+z^{2}}\). [Hint: Use the chain rule.]
(d) \(f=1 / r\).
Questions & Answers
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QUESTION:
Calculate the gradient \(\nabla f\) of the following functions, \(f(x, y, z)\):
(a) \(f=x^{2}+z^{3}\).
(b) \(f=k y\), where k is a constant.
(c) \(f=r \equiv \sqrt{x^{2}+y^{2}+z^{2}}\). [Hint: Use the chain rule.]
(d) \(f=1 / r\).
ANSWER:Step 1 of 5
(a)
It is known that \(\nabla f=\hat{\mathbf{x}} \frac{\partial f}{\partial x}+\hat{\mathbf{y}} \frac{\partial f}{\partial y}+\hat{\mathbf{z}} \frac{\partial f}{\partial z}\).
It is given that \(f=x^{2}+z^{3}\).
Find \(\frac{\partial f}{\partial x}\) as follows.
\(\begin{aligned}
\frac{\partial f}{\partial x} & =\frac{\partial}{\partial x}\left(x^{2}+z^{3}\right) \\
& =2 x
\end{aligned}\)
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